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Signal
conditioner
Data acquisition
module
Fixture
Computer
MIC1
MIC3
MIC2
Soybeans
Speaker
Tank
Fig. 1.
System of sound travel time measurement
4 Time-Delay Estimation Method Using Triple Correlation with
Wavelet De-noise
4.1 Cross-Correlation (CC)
Supposed,
x
1
(
n
)
and
x
2
(
n
) are the signals measured by location A and B respectively,
and they can be written in the form
xn
()
=+
sn
()
nn x n
() ,
()
=⋅
α
sn D
(
−+
)
n n
() .
(4)
1
1
2
2
where
s
(
n
) is the available signal
D
is time-delay of signal transmitted from A to
B
n
1
(
n
) and
n
2
(
n
) are additive noise
α
is the attenuation factor of signal.
The cross-correlation [5] of
x
1
(
n
)
and
x
2
(
n
) is expressed as
[
]
(
)(
)
R m Exnxnm
( )
=
() (
+
)
=
E sn nn
⎡
()
+
()
α
⋅
snmD nnm
(
+
−
)
+
(
+
)
⎤
⎣
⎦
xx
1
2
1
2
(5)
12
=⋅
α
RmD
(
− +⋅
)
α
R mD R m R m
(
− +
)
(
)
+
(
) .
ss
n s
sn
n n
1
2
1
2
According to the properties of auto-correlation, it can be known that |
R
ss
(
m-
D
)
|
≤
R
ss
(0). And
m
=
D
,
R
x
1
x
2
(
D
)=max[
R
x
1
x
2
(
m
)] is the time-delay estimation between
x
1
(
n
)
and
x
2
(
n
) with cross-correlation.
4.2 Cross-Correlation with Wavelet Transform (WT-CC)
Mallat decomposition and reconstruction algorithm of wavelet transform is used to
de-noise the signal of
x
1
(
n
)
and
x
2
(
n
). It includes three steps:
(1) Fix the wavelet mother function. Decompose the one-dimensional digital signal
with wavelet transform. In this paper, db18 is fixed as mother function of wavelet
based on experiment analyses. (2) Fix the scale of decomposition
j
.
If the bandwidth
of sampling signal is
f
k
, after Mallat decomposition in the scale of
j
, the low frequency
coefficient of signal
k
a
is 0~
f
k
/2
j
[6].
Therefore, if the upper limit of available bandwidth of signal is
f
, the scale
j
must be
appropriate to make
f
k
/2
j
approached to
f
. In this paper, the sampling frequency of the
system is 100kHz, according to Nyquist sampling theory, the bandwidth of sampling
()
j
is obtained, and the range of bandwidth of
()
j
a
k
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